What is Imaginary numbers: Definition and 78 Discussions

An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i2 = −1. The square of an imaginary number bi is −b2. For example, 5i is an imaginary number, and its square is −25. By definition, zero is considered to be both real and imaginary.Originally coined in the 17th century by René Descartes as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of Leonhard Euler (in the 18th century) and Augustin-Louis Cauchy and Carl Friedrich Gauss (in the early 19th century).
An imaginary number bi can be added to a real number a to form a complex number of the form a + bi, where the real numbers a and b are called, respectively, the real part and the imaginary part of the complex number.

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  1. BadgerBadger92

    B Tachyons and imaginary numbers?

    https://www.scientificamerican.com/article/what-is-known-about-tachy/ How do tachyons have mass of a square root of a negative number? I thought this was not mathematically possible?
  2. How Imaginary Numbers Were Invented

    How Imaginary Numbers Were Invented

    How Imaginary Numbers Were Invented [Veritasium]
  3. PainterGuy

    I Video on imaginary numbers and some queries

    Hi, I was watching the following video. I found some points confusing. Could you please help me to understand the gaps? Thanks, in advance! Question 1: Around 4:22, the video says the following. So for those mathematicians, negative numbers didn't exist. You could subtract, that is find...
  4. S

    B Why are imaginary numbers called "imaginary"? If they really exist

    If Imaginary numbers do exist and have real applications, then why do we call imaginary numbers "imaginary numbers"? . They exist. They're used all the time. What makes them "imaginary"?
  5. S

    Simplify this fraction containing imaginary numbers

    i can get to 3i+1/1-3i but no further. I take it this is the correct way to start
  6. F

    B Can Negative Numbers Be in the Domain of the Square Root Function?

    Consider the function sqrt(x). What is the domain of this function? Is it all real positive numbers? This is what I was taught in high school, but I was also taught that plugging in -1 would give an answer of i. So if the function takes negative inputs, shouldn’t they be part of the domain?
  7. Tan Thom

    MHB How do I solve for a_3 in a complex Fourier series?

    Good morning, I am working on a problem where I am finding the 4th Coefficient in a sample of 4 discrete time Fourier Series coefficients. I got the sum but now I have to solve for a_3 which consists of a real and imaginary part. Any assitance on how to solve for the a_3? Thank you. $a_k =...
  8. M

    B Simplifying roots of negative numbers

    In this Khan Academy video they say that it is ok to break the square root ##\sqrt{a\cdot b}##, with ##a, b \in \mathbb{R}##, into the product of two square roots ##\sqrt{a}\cdot \sqrt{b}##, only when: (1) both are non negative, (2) one of the two is negative and the other is possitive. I...
  9. I

    MHB Imaginary numbers and real numbers?

    z is either a real, imaginary or complex number, and z^12=1 and z^20 also equals 1. What are all possible values of z? I know 1 and -1 are them, and I think its also i and -i?
  10. Rectifier

    Finding anitderivative using complex numbers and Euler

    I have to find a primitive function below using the Euler formulas for ##\sin x## and ## \cos x## The problem $$ \int e^{2x} \sin 3x \ dx $$ Relevant equations ## \cos x = \frac{e^{ix}+e^{-ix}}{2} \\ \sin x = \frac{e^{ix}-e^{-ix}}{2i} \\ \\ \int e^{ix} \ dx = \frac{e^{ix}}{i} ## The attempt...
  11. FallenApple

    I What was the historical problem with Imaginary Numbers?

    I don't see why imaginary numbers were necessarily so difficult among top mathematicians back then. From pleano's axioms, we can derive the fact that any negative natural number times another negative natural number must be positive. Then this result extends to the reals, using theorems derived...
  12. M

    A If the axiom of induction were extended to include imaginary numbers....

    If the axiom of induction was extended to include imaginary numbers, what effect would this have? The axiom of induction currently only applies to integers. If this axiom and/or the well ordering principle was extended to include imaginary numbers, would this cause any currently true statements...
  13. AaronQ

    B Predicting outputs of f(x)=(1+i)^x

    I got bored a while back and deiced to create a table of the integer inputs of f(x)=(1+i)^x and I noticed quiet a few patterns which I am trying to catalog here, although most of my work so far deal with Natural inputs, all patterns continue into the negative, see here, I was wondering if anyone...
  14. N

    Mathematica Why Doesn't Chop[] Remove I.0 from the Output?

    As I found on many websites, they suggest to use Chop[]. I tried that already but it doesn't work. This is my output. {{2 (Conjugate[ I.0] I.0 + (Conjugate[I.0] + Conjugate[Subscript[v, 1]]/Sqrt[ 2]) (I.0 + Subscript[v, 1]/Sqrt[2])) Subscript[\[Lambda], 1] (Conjugate[I.0]...
  15. DaveC426913

    Explaining imaginary numbers to laypeople

    I've had discussions with laypeople (of which, I am one) about real-world manifestations of imaginary numbers. We can never seem to find a satisfactory, concise example. I know they are used in real-world calculations for things like EM wavelengths in electronics, but if you aren't into...
  16. thegirl

    Cross product imaginary numbers

    Hi, I was just wondering if you have a cross product can you multiply out the constants and put them to one side. So ik x ik x E is equal to i^2(k x k x E) therefore is equal to -k x k x E. Is that correct?
  17. S

    Exploring Imaginary Numbers for Beginners

    What are imaginary numbers? Does anyone know a good book for it?
  18. kostoglotov

    Question concerning 2nd order homogeneous linear diff eqs

    Homework Statement Regarding the case where the auxillary (characteristic) equation has complex roots, we solve the quadratic in the usual way using i to get the general solution y(x) = e^{\alpha x}\left(C_1 \cos{\beta x} + i C_2 \sin{\beta x}\right) And the textbook shows y(x) = e^{\alpha...
  19. 2

    Finding the nth root of a complex number?

    Homework Statement Find the solutions to z^{\frac{3}{4}}=\sqrt{6}+\sqrt{2}i Homework Equations de Moivre's theorem The Attempt at a Solution z^{\frac{3}{4}}=2\sqrt{2}e^{\frac{\pi i}{6}}=2\sqrt{2}e^{\frac{\pi i}{6}+2k\pi}=2\sqrt{2}e^{\frac{\pi +12k\pi}{6}i} z=4e^{\frac{4}{3}{\frac{\pi...
  20. G

    Feynman rules - where do the imaginary numbers come from?

    I'm trying to learn how to derive Feynman rules (what else to do during xmas, lol). The book I'm using is QFT 2nd ed by Mandl&Shaw. On p 428 they're trying to show how to derive a Feynman rule for W W^\dagger Z^2 interaction term g^2 \cos^2\theta_W\left[W_\alpha W_\beta^\dagger Z^\alpha Z^\beta...
  21. S

    Does (-2)^(⅔) have an imaginary component?

    Some calculators say (-2)2/3 is equal to ##-\frac{1}{2^\frac{1}{3}}+i\frac{3^\frac{1}{2}}{2^\frac{1}{3}}## while others say its equal to ##4^{\frac{1}{3}}## i.e. ##|-\frac{1}{2^\frac{1}{3}}+i\frac{3^\frac{1}{2}}{2^\frac{1}{3}}|##. I think I am right to imply from above that (-2)2/3 does have an...
  22. E

    Why Does i^2 Equal -1? Explained

    I see many people saying that the imaginary number squared is -1, like so: i = sqrt(-1) i2 = sqrt(-1)*sqrt(-1) = (sqrt(-1))2 = -1 But, what about this: i2 = sqrt(-1)*sqrt(-1) = sqrt(-1*-1) = sqrt(1) = 1 Can someone please explain to me why i2 = -1 if the above counter example is correct? If...
  23. H

    Nodal Analysis: Imaginary Numbers

    Homework Statement I have to find the Thevinin Equivalent for the following circuit. I am assuming the current is going out of the node. V= node between inductor and capacitor V0 = V[40/(40-150j)] (V-75)/(600+150j) + (-0.02V0) + V/(40-150j) = 0 The only problem I have is with the last...
  24. B

    Exploring the Properties of Vectors and Imaginary Numbers

    Homework Statement Consider a vector z defined by the equation z=z1z2, where z1=a+ib, z2=c+id. (a) show that the length of z is the product of the lengths of z1 and z2. (b) show that the angle between z and the x-axis is the sum of the angles made by z1 and z2 separately. The Attempt at a...
  25. P

    What inspired mathematicians invent imaginary numbers?

    Let me start by writing about the natural or counting numbers. The story of how, where and when we invented them is lost in the misty dawn of history. But perhaps our African ancestors, like our living simian cousins (and some other animals) evolved the ability to distinguish between few and...
  26. S

    Any real world use of imaginary numbers?

    Everybody says that it is used in engineering or somewhere but how can you use it. in real world it is impossible to take square of any number and get negative answer. how can it have any use when it does not even exist. and people talk about imaginary plane, what is it? Thanks for helping...
  27. ElijahRockers

    How Do You Prove the Inequality Involving Complex Numbers in Homework?

    Homework Statement Derive the following relation, where z1 and z2 are arbitrary complex numbers |z1z2*+z1*z2| ≤ 2|z1z2| The Attempt at a Solution I found the expression |z1z2*+z1*z2| = |2(a1a2+b1b2)| = √(4[a12a22 + 2a1a2b1b2 +b12b22]) But that is where I get stuck. How does the...
  28. E

    Question regarding imaginary numbers and euler's formula

    So, I was thinking about Euler's formula, and I noticed something interesting. Based on the fact that e^\frac{i\pi}{2} = 1 , it seems as though \frac{i\pi}{2} = 0. However, this doesn't make any sense. Not only can I not see how this expression could possibly equal 0, but that would imply that...
  29. H

    Imaginary numbers and the real part of the Schrodinger Equation

    At the moment I am studying the Schrodinger equation using this resource. In a 1D solution (sec 3.1 in the paper) they show that a wave function can be expressed as \Psi(x,t)=\sqrt{2}e^{-iE_{n_x}t}\sin (n_x\pi x) where n_x is the quantum number. And they show the real part of the solution in...
  30. M

    Simplifying with Imaginary Numbers

    Homework Statement I'm trying to see if what I have before the e match up with the correct answer. the correct answer is (2+.5i)e^(1+3i)x + (2-.5i)e^(1-3i)x The Attempt at a Solution This is what I have so far. I don't know how I would simplify anymore. Please help.
  31. H

    Question about imaginary numbers

    Homework Statement For a real number x, √√(-x) equals : a) +x b) -x c) complex d) pure imaginary Homework Equations √-1 = i The Attempt at a Solution Here is what i did: If x is a positive real number then the answer comes out to be x^0.25 * √i (now what is square...
  32. M

    Complex / imaginary numbers trigonometric graphs

    Hi, I have been representing complex numbers in graphical form in school recently. My teacher was telling me about a graph which shows all 4 quadrants and basically shows you what each quadrant is in terms of pi. Hopefully you understand what I mean, I have been looking on the internet for this...
  33. R

    Imaginary numbers concept help

    I've been learning about imaginary numbers and while I understand the concept of them I have tried a few examples with them and I don't get some of the answers. why can you not take xi = 90i and multiply it by i x*i*i = 90*i*i x*-1 = 90*-1 -x=-90 x=90 Thanks AL
  34. N

    Are imaginary numbers really necessary?

    You guys are probably sick of people who know little math posting here, but there's something that's been bugging me. I've bought The Feynman Lectures on Physics and have been reading through it slowly, and I'm up to the part where he talks about probability amplitudes of the electrons/photons...
  35. D

    Resistor Heat Dissipation (with imaginary numbers)

    Friends: I am wondering about heat dissipation when you have imaginary numbers. Lets say a current I = (3 + 4j) Amps is going through an impedance Z = (2 + 3j) Ohms. What is the amount of heat dissipated by the impedance? I think that you take the magnitude of the current, |I| = 5 Amps, and...
  36. L

    Imaginary numbers what to do with them

    I have attached an image showing the three possible solutions (as determined by Mathematica) when solving for the peak velocity(Vs) in a trapezoidal move where the following are already known: distance(d),total time(t),units of acceleration(Ma),units of deceleration(Ma),initial velocity(Vi)...
  37. J

    Simplification with exponentials and imaginary numbers

    Hi, I'm using euler's identity : exp(i∏) = exp (-i∏) = -1 to simplify the equation after integrating it. [PLAIN]http://img443.imageshack.us/img443/5504/captureikm.jpg Note: the equation to be integrated is exp(0.5it) + exp(-0.5it) and they have simplified it, it was actually a cos(0.5t)...
  38. L

    Imaginary numbers multiply and divide

    I don't really understand how to multiply and divide when numbers are in a+bi form
  39. vibhuav

    Physical meaning of imaginary numbers

    Can someone give a physical meaning for imaginary numbers? The imaginary numbers, in my opinion, are truly imaginary. What do they even represent? Irrational numbers are, well, preposterous but I can accept them. √2, π and φ have some tangible meaning, but √(-1)? What does it mean? A solution...
  40. A

    Imaginary Numbers: My Number System & Research Paper

    Well I have developed a number system which allows the existence of imaginary numbers. Please visit it at : http://www.scribd.com/doc/46064105/Math-Paper. An intro of these ideas is presented at :http://www.scribd.com/doc/46117043/Introduction-to-My-Research-Paper Please provide me feedback...
  41. N

    Exponents and Imaginary Numbers

    Hello, I did the integral of a Fourier Transform which resulted in this: A(je^(-jwe^(To+t/2) - je^-jw(T0-t/2))/(1/w) Where A is the amplitude, j the imaginary number, and w is omega or 2*pi*f. My question is, how can this be further simplifier. I am forgetting how to simplify...
  42. V

    Solve Imaginary Numbers: 2*EXP(i*pi/3) -> 1+sqt(3)i

    Homework Statement Write 2*EXP(i*pi/3) in the form \alpha + i\beta Answer is given = 1 + sqt(3)i Homework Equations The Attempt at a Solution I'm supposed to turn this exponential form of imaginary number into a standard form in order to solve an ODE. I have no idea how they got 1+sqt(3)i...
  43. F

    Are there imaginary numbers other than i?

    Are there "imaginary" numbers other than i? I'm taking a class in complex analysis and the professor wrote the textbook so I'm getting most of it. There is one elephant in the room though, and I haven't been able to make office hours to clear it up. Are there "imaginary" numbers other than...
  44. C

    Imaginary Numbers and Properties: A Puzzling Case

    -1/1=1/-1 sqrt(-1/1)=sqrt(1/-1) i/1=1/i i*(i/1)=i*1/i i^2/1=i/i -1/1=1 -1=1 <-- Well my conclusion is that properties don't work with imaginary numbers or did i do something wrong?
  45. C

    Dividing/Multiplying Imaginary Numbers

    Is there a more convenient way to multiply and divide imaginary numbers than converting back and forth from phasors? (I guess I should say "when also having to add and subtract them") I always find AC circuit calculations to be tedious and problem filled when I do it that way. For example...
  46. T

    Imaginary Numbers in a general homogenous solution for a differential equation

    Homework Statement Find the general solution for: y''+2y'+5y=3sin2t The attempt at a solution y''+2y'+5y=3sin2t First step is to find the general solution to the homogenous equation, so skipping 2 steps (letting y=e^rt and dividing) R^2+2r+5 (-2+/- sqroot(4-4*5))/2 =-1 +/- 2i...
  47. genphis

    Imaginary numbers negative confusion

    i know this must seem real stupid but if 1 x 1 =1 ( square root wise) how can -1x-1=+1 again square root wise. i am reading fermats last theorum to me if you times negative you increase the negative. i don't see why the imaginary numbers need to be invoked. i understand the argument for...
  48. T

    Imaginary Numbers to Polar form

    Homework Statement (1+i)i = reiθ Find the real values of r and θ. The Attempt at a Solution Well, after doing a similar(ish) question I decided taking logs would be a good start: i loge(1+i) = loger + iθ From here, I have no idea where to go. Using a power of i is killing me...
  49. S

    Implications of Imaginary Numbers?

    Hello, I have a quick question that I imagine anyone who has studied physics or math at a university can answer rather easily. If not, I apologize in advance for the effort! What is the physical significance of imaginary numbers? I have heard repeatedly that imaginary numbers are relevant...
  50. T

    Imaginary numbers and Imaginary Time

    Imaginary numbers are a lot less mysterious than they sound. They are the result from trying to take the square root of a negative number. They are called “imaginary” because they don’t exist in the normal number system, normally you can’t take the square root of a negative number because the...